FACULDADE DE CI ˆENCIAS ECON ˆOMICAS PROGRAMA DE P ´OS-GRADUAC¸ ˜AO EM ECONOMIA
CHANT ´OS GUILHERME ANTUNES MARIANI
ESSAYS ON FREEDOM OF CHOICE AND CAPABILITIES
Porto Alegre 2019
ESSAYS ON FREEDOM OF CHOICE AND CAPABILITIES
Tese submetida ao Programa de P´os-Graduac¸˜ao em Economia da Faculdade de Ciˆencias Econˆomicas da UFRGS, como quesito parcial para obtenc¸˜ao do t´ıtulo de Doutor em Econo-mia, com ˆenfase em Economia Aplicada.
Orientador: Prof. Dr. Flavio Vasconcellos Comim
Porto Alegre 2019
Mariani, Chantós Guilherme Antunes Mariani Essays on Freedom of Choice and Capabilities / Chantós Guilherme Antunes Mariani Mariani. -- 2019. 105 f.
Orientador: Flavio Vasconcellos Comim Comim.
Tese (Doutorado) -- Universidade Federal do Rio Grande do Sul, Faculdade de Ciências Econômicas, Programa de Pós-Graduação em Economia, Porto Alegre, BR-RS, 2019.
1. Freedom of Choice. 2. Capabilities. 3. Experimental Economics. 4. Conjoint Analysis. I. Comim, Flavio Vasconcellos Comim, orient. II. Título.
Elaborada pelo Sistema de Geração Automática de Ficha Catalográfica da UFRGS com os dados fornecidos pelo(a) autor(a).
ESSAYS ON FREEDOM OF CHOICE AND EXPERIMENTAL ECONOMICS
Tese submetida ao Programa de P´os-Graduac¸˜ao em Economia da Faculdade de Ciˆencias Econˆomicas da UFRGS, como quesito parcial para obtenc¸˜ao do t´ıtulo de Doutor em Econo-mia, com ˆenfase em Economia Aplicada.
Aprovada em: Porto Alegre, 26 de abril de 2019
Banca Examinadora
Prof. Dr. Flavio Vasconcellos Comim - Orientador Universidade Federal do Rio Grande do Sul (UFRGS)
Prof. Dr. Ely Jos´e de Mattos
Pontif´ıcia Universidade Cat´olica do Rio Grande do Sul (PUCRS)
Prof. Dr. Izete Pengo Bagolin
Pontif´ıcia Universidade Cat´olica do Rio Grande do Sul (PUCRS)
Prof. Dr. Sabino da Silva Pˆorto J´unior
A presente tese desenvolve trˆes ensaios que abordam diferentes temas relacionados `a liber-dade de escolha. No primeiro ensaio, propomos uma regra para ranquear conjuntos de oportu-nidades conforme a liberdade de escolha que eles propiciam, e que leva considerac¸˜ao as meta-preferˆencias dos indiv´ıduos. Desenvolvendo um abordagem te´orica, investigamos se, ao con-siderarmos indiv´ıduos com m´ultiplos objetivos, algumas noc¸˜oes usuais acerca liberdade que foram propostas na literatura s˜ao modificadas. Os resultados mostram que a regra proposta vi-ola o axioma da monotonicidade, e que indiv´ıduos podem atribuir maior liberdade de escolha a conjuntos com menos opc¸˜oes. No segundo ensaio, propomos um experimento online baseado em an´alise conjunta para avaliar como a liberdade de escolha dos indiv´ıduos ´e afetada pelas caracter´ısticas dos menus que os agentes disp˜oem no momento de realizar escolhas. Estudamos o efeito de trˆes bases informacionais propostas na literatura – a cardinalidade dos conjuntos, a diversidade das opc¸˜oes, e a qualidade dessas opc¸˜oes – e tamb´em investigamos se normas de comportamento social podem influenciar a liberdade de escolha dos agentes. Usando es-timadores de diferenc¸as em m´edias operacionalizados atrav´es de uma ´unica regress˜ao linear, os resultados sugerem que n˜ao apenas os elementos tradicionais como a cardinalidade, diver-sidade, e preferˆencias, s˜ao fundamentais para entender liberdade de escolha, como tamb´em normas de comportamento social tem impacto significativo. Finalmente, no terceiro ensaio, investigamos como indiv´ıduos realizam trade-offs entre diferentes dimens˜oes de suas vidas quando consideram os seus potenciais para atingir melhores n´ıveis de bem-estar. Realizando um novo experimento baseado em an´alise conjunta, estudamos seis dimens˜oes da vida dos in-div´ıduos que s˜ao considerados centrais pela abordagem das capacitac¸˜oes, e avaliamos como diferentes circunstˆancias em cada uma dessas dimens˜oes afeta a liberdade de bem-estar dos indiv´ıduos. Os resultados mostram que as dimens˜oes relacionadas `a seguranc¸a dom´estica e a boas acomodac¸˜oes s˜ao de grande relevˆancia, e, portanto, os formuladores de pol´ıticas p´ublicas podem encontrar espac¸o para o estabelecimento de prioridades. Usando um modelo hier´arquico Bayesiano, tamb´em investigamos se a importˆancia que os participantes d˜ao `as dimens˜oes estu-dadas varia conforme o n´ıvel de bem-estar que eles reportaram no estudo, e tamb´em se varia conforme a situac¸˜ao de vida em cada uma dessas seis dimens˜oes. Os resultados sugerem que indiv´ıduos que indicam menor bem-estar atribuem menor importˆancia para a dimens˜ao da vida relativa ao amor dos familiares pr´oximos se comparado a outros participantes com maior n´ıvel de bem-estar subjetivo. Tamb´em, os participantes que indicaram ter sofrido alguma forma de violˆencia dom´estica no passado veem de forma menos negativa a ausˆencia de uma boa condic¸˜ao de seguranc¸a dom´estica, se comparado aos indiv´ıduos que nunca sofreram com esse tipo de violˆencia.
This thesis presents three essays that approach different topics related to freedom of choice. In the first essay, we propose a rule to rank opportunity sets in terms of freedom of choice that considers information about individuals meta-preferences. From a theoretical perspective, we investigate whether accounting for a person’s multiple goals and objectives changes some com-mon notions about freedom of choice that have been proposed by the literature. We show that our rule fails to respect the monotonicity axiom, and that individuals might experience greater freedom when some options are excluded from their initial opportunity sets. In the second es-say, we propose an online conjoint experiment to evaluate how individuals’ freedom of choice is affected by the characteristics of the menus they have at their disposal at the moment of choice. We study the effect of traditional informational basis used to evaluate freedom of choice – sets’ cardinalities, the diversity, and the quality of their options – and also investigate whether social norms of behavior have some bearing on freedom. Using a difference-in-means estimator that is operationalized through a single linear regression, our results suggest that not only traditional elements such as cardinality, diversity, and preferences are key to understand freedom, but also that social norms of behavior have a significant impact. Finally, in our third essay, we inves-tigate how trade-offs between capabilities take place when individuals consider their ability to achieve higher well-being. Using another conjoint experiment, we study six life domains that are considered as central by the capability approach, evaluating how varying the situation of in-dividuals in each of these domains affect their well-being freedom. Our results show that being secure from domestic violence and enjoying a decent shelter have great relevance, and hence policymakers might find room for establishing priorities. Using a hierarchical Bayesian model, we also investigate whether the relative importance that people give to these life domains vary with participants’ subjective well-being, and with participants’ self-reported situations in each of the life domains studied. We find that subjects that reported lower well-being attached less importance to enjoying the love and care of their families as compared to those participants with higher subjective well-being. Moreover, respondents that reported to have suffered from some sort of domestic violence find less harmful the lack of a decent level of bodily security, as compared to those participants that have never suffered from such type of assault.
Figure 3.1 – Example of a conjoint table seen by participants during the experiment 43
Figure 3.2 – AMCEs (OLS estimates) . . . 49
Figure 3.3 – Screenshot of the testing phase of the experiment . . . 62
Figure 3.4 – Screenshot of the conjoint phase of the experiment . . . 63
Figure 4.1 – Effects of attributes (subjective well-being) . . . 79
Figure 4.2 – Effects of attributes (levels of capabilities) . . . 80
Figure 4.3 – An example of choice task from the conjoint phase . . . 88
Figure 4.4 – Traceplots of AMCEs (heterogeneity in subjective well-being) . . . . 92
Table 3.1 – Descriptive statistics of participants . . . 44
Table 3.2 – Attributes and values used in the conjoint experiment . . . 45
Table 3.3 – AMCEs (OLS estimates) . . . 51
Table 3.4 – Model breakout by gender: AMCEs (OLS estimates) . . . 54
Table 4.1 – Levels of attributes used in the conjoint stage . . . 70
Table 4.2 – Descriptive statistics of participants . . . 72
Table 4.3 – Distribution of groups per dimension and gender . . . 76
Table 4.4 – AMCEs (OLS estimates) . . . 78
Table 4.5 – Summary of the posteriori distribution of βji (subjective well-being) . 82 Table 4.6 – Summary of the posteriori distribution of βji (levels of capabilities) . . 83
Table 4.7 – Survey (Part 1) . . . 89
Table 4.8 – Survey (Part 2). . . 90
1 INTRODUCTION . . . 8
2 MULTIPLE GOALS, META-PREFERENCES, AND FREEDOM OF CHOICE COMPARISONS . . . 11
2.1 INTRODUCTION . . . 11
2.2 FRAMEWORK . . . 17
2.3 PROPERTIES OF OPTIONS IN CR2R1(A, B) . . . 20
2.4 RESULTS . . . 23
2.5 CONCLUDING REMARKS . . . 30
2.6 REFERENCES . . . 30
3 SOCIAL APPROPRIATENESS AND FREEDOM OF CHOICE: A CON-JOINT EXPERIMENT . . . 34
3.1 INTRODUCTION . . . 34
3.2 SOCIAL NORMS, BEHAVIOR AND FREEDOM . . . 36
3.3 EXPERIMENTAL DESIGN AND METHODOLOGY . . . 40
3.3.1 Conjoint experiments . . . 41
3.3.2 Our experimental design and sample . . . 42
3.3.3 Estimation strategy . . . 46
3.4 RESULTS . . . 48
3.4.1 Results: Pooled sample . . . 48
3.4.2 Heterogeneous effects: Gender . . . 50
3.4.3 Discussion and policy implications . . . 52
3.5 CONCLUDING REMARKS . . . 55
3.6 REFERENCES . . . 56
3.7 APPENDIX . . . 62
4 ASSESSING THE RELATIVE IMPORTANCE OF DIFFERENT LIFE DO-MAINS FOR WELL-BEING FREEDOM . . . 64
4.1 INTRODUCTION . . . 64
4.2 OUR EXPERIMENTAL DESIGN AND SAMPLE . . . 68
4.3 METHODS . . . 73
4.4 RESULTS . . . 75
4.4.1 Groups . . . 75
4.4.2 Overall AMCEs . . . 76
4.4.3 AMCEs: Breakouts by participants’ types . . . 81
4.5 CONCLUDING REMARKS . . . 82
4.6 REFERENCES . . . 84
4.7 APPENDIX . . . 88
5 CONCLUDING REMARKS . . . 94
1 INTRODUCTION
The instrumental importance of freedom of choice – when freedom is seen as means to given ends – is widely recognized by traditional economic analysis, while its intrinsic impor-tance – when freedom is an end in itself – is an aspect somewhat neglected (SEN, 1988). The usual ‘story’ depicts a consumer endowed with preferences that can be represented by a quasi-concave utility function, and whose choices are made with the goal in mind to maximize the value of this function. For this ‘utility-driven person’, whose multiple cares and objectives are perfectly described by an all-things-considered preference ordering, the only relevant informa-tion throughout the process is which of the feasible alternatives gives the highest utility level, with no specific importance attached to the unchosen options that she has faced, or to the pro-cess of choice. For instance, if an option x maximizes the agent’s utility, under a pure utilitarian perspective there cannot be any difference between choosing x from an opportunity set that only contains x, and a situation where x is chosen from a set with a large variety of options.
Nevertheless, even if x is considered the best option, it can be reasonably argued that some-thing valuable is lost, and that opportunities are severely curtailed when choice is constrained to the singleton set {x}. In other words, the agent might also attach importance to what she could havechosen, or to the possibility of deciding differently, making the act of choosing x from a larger set more valuable than the same choice from a set that only offers a single alternative to the agent. This line of inquiry motivates what became known as the ‘Freedom of Choice literature’, that aims to address freedom of choice not only through its instrumental importance i.e. people’s ability to choose freely for themselves, without constraints imposed by others -but by its intrinsic importance to decision-makers (GAERTNER; XU, 2011)
Among the exponents of this literature are names such as Prasanta Pattanaik, Yongsheng Xu, Amartya Sen, Wulf Gaertner, Clemens Puppe and many others that have largely contributed to Social Choice theory. Indeed, the freedom of choice literature tends to be a highly mathemat-ical research field that draws heavily on social choice methodologmathemat-ical tradition to establish its main results. This line of research has also a close connection with Amartya Sen’s and Martha Nussbaum’s ‘Capability approach’ to human development, which sees the expansion of peo-ple’s freedoms - broadly understood as the set of what one is able to do and to be - as the main objective of development policies. Therefore, the freedom of choice literature contributes largely to distinct realms of research, and might motivate both theoretical and empirical studies. Within this literature, there is ample variation regarding what information should be consid-ered to establish comparisons in terms of freedom of choice. One aspect that has been receiving attention relates to preferences among options and what role - if any - should they play. Amartya Sen favored the view that preference information cannot be disregarded, suggesting that any ax-iomatic structure intended to reflect judgments of freedom of choice should not only account for the number of alternatives - the quantitative aspect - but also encompass the qualitative
in-formation provided by preferences. For instance, if a person feels that all the options within a set A are outstanding, while all alternatives in a set B are terrible, this qualitative information should influence the way in which A and B are ranked in terms of freedom. Many authors followed this same reasoning and proposed rankings of opportunity sets in terms of freedom of choice where preferences are taken into account (SEN, 1991; PATTANAIK, XU, 1998; PUPPE, XU, 2010).
There are many aspects, however, that remain unexplored by this literature, and motivate this research. The first aspect refers to what Amartya Sen called ‘rankings of preference rank-ings’, or meta-rankings, in his insightful ‘Rational Fools’ paper (SEN, 1977). An analytical structure based on meta-rankings allows the expression of moral judgments, or, more generally, may be used to describe how distinct preference orderings - each motivated by some specific ‘rationale’ that represents a feature of the agent’s plural identities - reflect values and principles that a person has reasons to treasure during her life. Enriching the analytical structure so as to incorporate meta-preferences would make possible the tractability of a new range of elements, expanding the reach of the theoretical discussion of freedom, and bringing a new set of insights to our understanding of what elements can enhance freedom when it is seen as opportunities for choice.
Thus, in our first essay, we propose a preference-based rule to rank opportunity sets in terms of freedom of choice. In our framework, agents are endowed with preferences over alternatives, and preferences over distinct preference orderings that reflect their ‘multiple-selves,’ what de-notes a situation in which information about meta-preferences is also relevant to freedom of choice comparisons. After we axiomatic characterize our rule, we study under which condi-tions this rule respects some desired properties such as the ‘Monotonicity axiom,’ where we show that – when a richer structure of preferences is considered – freedom may decrease when one option is added to the existing menu.
A second aspect that is worth noticing and motivates this thesis is the lack of empirical re-search in this field. So far, the majority of studies on freedom of choice concentrate in exploring the axioms, and which rankings can be constructed based on these axioms, but little has been done in designing experiments to test the theoretical predictions. In this sense, this research comes as an attempt to reduce this gap, and provide new evidence about the determinants of freedom of choice.
In our second essay, therefore, we propose an online conjoint experiment to evaluate what characteristics of an opportunity set makes it better evaluated in terms of freedom. In addition to three usual features of sets – cardinality, diversity of options, and the quality of possible alterna-tives considering agents’ preferences – we suggest that social norms of behavior also may play a significant role in peoples’ freedom to choose. Our hypothesis is that people might enjoy less freedom of choice when they find themselves in situations where a trade-off between choosing their best options, and doing what is regarded as more socially appropriate by others, occurs. Compared to a baseline set, our results suggest that sets with fewer options tend to receive less
support in terms of freedom, while the diversity of the alternatives does not necessarily enhance the freedom experienced by agents. Sets with low-quality options also provide less freedom than a baseline set with neither good nor bad options. Finally, our findings suggest that social norms of behavior may have a large influence on freedom of choice: on average, when other people rate an agents’ best choice in a menu as very socially inappropriate, this set becomes 7.9 percentage points less likely to provide greater freedom than another set with a socially neutral best choice. Also, our findings suggest that women and men might respond differently to the factors that impact their freedom to choose, with women, for example, being more sensitive to the influence of social norms of behavior than men.
Finally, in our third essay, we explore the notion of well-being freedom (SEN, 2009), and its relation to the different life domains that are cherished by individuals. More specifically, we use an online conjoint experiment to evaluate what dimensions of life are more relevant to individuals when taking into consideration a perspective based on their ability to pursue their own well-being. We study six different areas of life that are related to the central capabilities list in Nussbaum (2000, 2011), and operationalized using the survey instrument developed in Anand et al. (2009). Our results suggest that being secure from domestic violence, and being able to enjoy adequate shelter, are two factors extremely valued by the participants in our ex-periment, and hence should receive some priority by policymakers that are unable to deliver a full set of capabilities to all citizens in every dimension of life. Furthermore, we also investi-gate whether the importance that people attach to these dimensions vary with their subjective well-being levels, and with their conditions in each of these life domains. Our results indicate that the results are robust to variations in these aspects in four of the six capabilities studied. In the dimension related to subjects’ bodily integrity, participants that have experienced domestic violence at some point in life attached less importance to the absence of a decent level of do-mestic security, as compared to those that have always enjoyed a decent level of this capability. Also, improving the capability related to family love did not have a positive effect on respon-dents that reported low subjective well-being, while such improvement affected significantly high subjective well-being participants.
The complexity of the everyday challenges requires from academia not only the proposal of novel theoretical approaches, but also the employment of new methodologies that may be found outside the usual toolkit of economists. This research takes some steps in these directions, and also reinforces the idea that expanding research on freedom of choice, opportunities and human behavior may contribute vastly to a better understanding of what are individuals’ priorities, which elements impose restrictions on the ability to choose, as well as shed some light on the human development debate.
2.1 INTRODUCTION
The opportunity aspect of freedom has been forcefully defended by Sen (1991, 1993b, 2009) as a broader and more adequate informational basis for social welfare evaluations. More free-dom, Sen argues, is associated with the expansion of people’s capability sets - the set of all functionings(i.e., all ‘doings’ and ‘beings’ of an individual) - what enhances people’s ability to promote whatever goals and objectives they have set to themselves. In this sense, expanding individuals freedoms becomes the primary means to the promotion of human development: an objective that draws heavily on the evaluative exercise of comparing different opportunity sets in terms of freedom of choice.
Many different rules have been proposed to tackle the issue of making freedom comparisons. If functionings - and not resources - are able to increase people’s freedom of choice, these rules can be simply interpreted as a way to compare two sets of options A and B in terms of freedom given a particular manner of converting resources into functionings. To illustrate the existing approaches, suppose the following situation: Mary, the older of two sisters, receives two notes of $50 from her parents and is instructed to think about giving some of this money to her younger sister, Jill. ‘It is completely up to you,’ says her mother. Assume further that Mary cannot exchange the $50 notes into smaller ones, leaving her options x =‘take all the money for herself’, y =‘give to Jill $50’, and z =‘give to Jill $100’. The set A = {x, y, z} denotes Mary’s opportunity set in this situation. Now, assume another situation where she receives the same instructions but, instead of receiving two notes of $50, Mary receives only a $100 note, leaving her with the opportunity set B = {x, z}. What scenario - A or B - offers to Mary larger opportunity-freedom?
The seminal paper of Pattanaik and Xu (1990), for instance, propose a rule where no in-formation other than sets’ cardinalities is necessary to make freedom comparisons. Under Pat-tanaik and Xu’s cardinality rule, the underlying assumption is that each resource is converted automatically into a different functioning and is capable of expanding an agent’s capability set. In Mary’s example, the set that gives larger freedom of choice is therefore obvious: given that A has three options while B has only two, the set A provides more freedom of choice than B.
However, it is not difficult to conclude that the cardinality rule fails to accommodate a variety of situations where the quantitative aspect is not the only one that seemingly plays a part. For instance, the diversity, or dissimilarity among options and what effect they have on freedom appraisals have been a widely pursued line of inquiry by a large body of the literature, where the intuitive notion that expanding a set with an option that is similar to already available options should do little to the expansion of the decision maker’s freedom has been addressed
(see, among others, PATTANAIK, XU, 2000; VAN HEES, 2004; PERAGINE, ROMERO-MEDINA, 2006; BERVOETS, GRAVEL, 2007; GUSTAFSSON, 2010).1
In another branch of the critiques to the cardinal approach, Sen (1991, 1993b) argues that relying solely on cardinality information makes one fail to depict clearly the opportunity aspect of freedom. For instance, it is counterintuitive to attach the same freedom of choice to two sets only because they have the same cardinalities even acknowledging that one of these sets have much more desirable options than the other. Therefore, any axiomatic structure intended to reflect a freedom ranking should not only account for the number of alternatives, but also expand the informational basis with agent’s preferences among these alternatives. For example, if Mary strictly prefers donating $50 over any other option because, say, she has always shared her possessions equally with her sister and feels that this is what is the right thing to do, any freedom ranking should be sensible to the fact that B does not allow her choosing what is best. Enriching the model with information about preferences, therefore, changes the manner in which resources are being converted into functionings and also makes room for the idea that each functioning contributes differently to the freedom of choice.2
In general, incorporating notions of similarity or preferences has been the usual response to Pattanaik and Xu’s pure cardinal approach (VAN HEES, 2004; DOWDING, VAN HEES, 2009), with few attempts to address jointly the two aspects (e.g., PERAGINE, ROMERO-MEDINA, 2006). In what concerns the approaches based on preferences, the question of what prefer-ences should count has been discussed forcefully, where the ‘reasonable preferprefer-ences’ argument of Jones and Sugden (1982), and further developed in Pattanaik and Xu (1998) and Romero-Medina (2001) has gained prominence. Basically, in order to assess whether an option is sig-nificant to increase freedom, one should examine the information given by those “preference orderings that a reasonable person in the agent’s situation can possibly have” (PATTANAIK, XU, 1998, p. 180). Thus, what matters is not the set of preferences actually held by the in-dividual, but a set of preference rankings that are regarded as ‘reasonable’ given the situation the agent finds herself in. To illustrate, adding an opportunity d = ‘burn all the money’ to ei-ther A or B would do little to enhance Mary’s freedom since this alternative is (presumably) unreasonable from the viewpoint of a reasonable agent.3
The argument based on reasonability allows the consideration of a broad range of pref-erences into the analysis. First, a particular ordering of alternatives may be reasonable even though the agent attaches zero probability to the possibility of effectively ranking alternatives
1To illustrate this fragility, Pattanaik and Xu (1990) pointed out that their cardinality-based rule leads to consider sets such as {train, red car} and {blue car, red car} as equivalent in terms of freedom, regardless the fact that the former apparently gives the decision-maker more variety than the latter.
2Note that preferences in this example follow the broader sense of preferences driven not only by self-interested motivations, as discussed in Sen (1993b). This conceptualization of preferences is rooted in the distinction between ‘tastes’ and ‘values’ proposed in Arrow (1950).
3Clearly, one critique that may arise to this sort of conclusion is that is always possible to find a situation where even an option such as ‘burning all the money’ is reasonable (for instance, Mary may burn the money to protest against her parents) and therefore contributes positively to freedom. For this and other critiques to the reasonableness approach, see Van Hees and Wissenburg (1999).
in that way, departing from the approach focused on preference for flexibility or uncertainty re-garding future tastes (see, among others, ARROW, 1995). In our example, based on the notion of reasonability, Mary can consider as reasonable the option of giving away all the money to her sister even though she has no plans to do it whatsoever. Second, the claim that alternatives increase freedom on the grounds of reasonability allows admitting that there may be a vast plu-rality of reasons that can be regarded so as to assess what preferences should count, what makes it an example of a multi-preference approach to freedom (PUPPE, XU, 2010).
Nonetheless, as pointed out in Jones and Sugden (1982), reasonability as the only required standard is incapable - or even do not intend to - rule out preference orderings upon which judgments of value can be made, such as moral judgments: the reasonableness criterion would, at most, exclude those rankings that are patently unjustifiable even using mild standards of reasonability. Moreover, Pattanaik and Xu (1998) do not specify which criteria is adequate to distinguish preference orderings into reasonable and unreasonable, or how reasonability acts as a screening device to ascertain which opportunities are relevant to the individual’s goals and ends she aims to promote.
In this sense, the richer structure of meta-preferences proposed in Sen (1977) may be helpful to give a more robust response to how various distinct preference orderings should be consid-ered. Sen (1977), in his strong critique of the narrow focus of standard decision theory, has advocated in favor of an analytical framework that incorporates the information given by meta-preferences to the analysis of what constitutes a rational and consistent choice pattern. The narrow structure of economic models, he argues, when attempts to synthesize a plurality of preferences into a unique complete, transitive and all-things-considered preference ordering, ends up impoverishing the analysis and neglecting essential aspects of the decision-making process such as the vast variety of principles, values, and reasons through which individuals can evaluate the same decision problem. An agent preoccupied solely with the rationality of her acts, relying exclusively on an all-things-considered preference and making no room to the plurality of principles that reflect her multiple goals is what Sen has defined as a ‘rational fool.’ To illustrate, take the example of Sen (1977) based on morality. Suppose that preferences R1
and R2 reflect, respectively, rankings of alternatives based on personal welfare when sympathy
is taken into account and when sympathy is disregarded, and a third preference ranking R3 that
describes actual choices made by the agent. By definition, these three rankings are reasonable since there are ‘reasons’ to adopt any of them as the criterion to make decisions. But assume further that this agent has a goal in mind of choosing the most moral actions because, say, the choice based on morality promotes the ends that she cherishes. In this case, one should expect that how these rankings compare to a hypothetical ‘most moral’ preference R4is an information
of relevance to freedom comparisons. Pattanaik and Xu’s (1998) preference-based rule makes room for the idea that an all-things-considered preference ordering may not suffice to account for the many rationales that can be appropriate to assess situations.4 However, when attaches 4Important to note, though, that nothing avert the individual from regarding only one preference ranking as
rea-equal relevance to each preference ordering regarded as reasonable, as if each could reflect with the same intensity one’s goals and values, the individual depicted by Pattanaik and Xu’s approach neglects information that might otherwise be important for an assertive comparison of sets in terms of the opportunity aspect of freedom.
Mary’s dilemma, for instance, depicts a conflict between two distinct points of view whose choice prescriptions can be rationalized by self-interested and other-regarding preferences; a conflict that has been extensively studied by behavioral economists through the experiment known as the dictator game (see Forsythe et al., 1994, and for a review of results, see also Engel, 2011). Back to our example, if we assess her decision problem through these two view-points, we have no difficulty to come up with a set of reasonable preference orderings that leads to the conclusion that all options are equally reasonable and capable of enhancing freedom: choosing x is certainly a nice possibility when Mary thinks exclusively about her own well-being, while y (or z) emerge as best options when she takes into account her sister’s situation. However, if individuals have ‘preferences over preferences’ based on how these reflect their plural identities, Mary may attach distinct importance to each principle driving these preference orderings, according to the extent that each reflects her plural interests and goals that she aims to promote. It could be the case that, even though she values the possibility of proposing an egalitarian allocation, Mary gives larger importance to her own well-being and enjoys larger freedom of choice when faces a menu that allows her choosing x even acknowledged that y and z are reasonable.
Regardless of what informational basis is used, a common feature of diversity-based or preference-based approaches is that freedom of choice cannot be decreased by the addition of new alternatives: a property commonly called as monotonicity (PUPPE, 1996; PUPPE, XU, 2010), or set dominance (SEN, 1991). In the framework proposed in Gravel (1994), for instance, comparisons in terms of freedom rely entirely on the notion of monotonicity. Sen (1991) argues that ‘if a person has an enlarged set to choose from, then his freedom, in some sense, must be at least as large, if not larger, no matter what his preferences are and what he actually chooses from the respective sets” (SEN, 1991, p. 21).
Nevertheless, this conclusion may be deceptive when we consider an agent that pursuits multiple goals simultaneously. For instance, imagine the situation of an individual that must pick one fruit from a fruit basket with an apple (a1) and a mango (m1).5 To this agent, both
fruits are enjoyable, but m1 is regarded as tastier than a1. Further, assume that we can add
either another mango (m2) or apple (a2) to the basket: if we desire to enhance the opportunity
freedom of this agent, what fruit should we put into the basket?
One plausible response to this query relies on the information about similarity: both m2 sonable. For example, when facing the menu ‘living healthy’ and ‘living unhealthy,’ one can have a hard time to rank the latter option above the former in any preference ordering based on reasonability standards. In this case, the set of all reasonable preference orderings over these two alternatives would be a singleton.
5This example is based on Sen (1993a) and his critique of the imposition of internal consistency conditions on choice functions.
and a2 add little to the agent’s freedom because they do not increase the diversity of options.
We argue, however, that we must assess what goals and objectives this agent aims to promote. If this individual aims only to pick the fruit that is best for her, it is possible to reason that, while having apples is relevant when we take into account the intrinsic importance of freedom, having mangoes is fundamental to her opportunity freedom because this is the type of fruit that she likes the most. Therefore, when taking into consideration this specific situation, it can be argued that, in the less favorable scenario, expanding the set with m2 or a2 at least will not
reduce this agent’s opportunity freedom.
Suppose, however, that this decision-maker desires not only pick her most preferred fruit, but also do it in a way that she will not appear as a selfish person to others, being the latter objective her primary goal. One possible way of not appearing selfish to others in this decision problem is, whenever possible, leave the basket with at least two different types of fruits. In this case, constraining her choice to the set {m1, a1, a2} imposes to her a trade-off between these
two goals: either she acts self-interestedly and chooses m1, abandoning her first goal, or she
picks one apple and leave some diversity to the basket, but refrains from fulfilling her craving for mangoes. In any case, the opportunity freedom - seen as freedom to achieve multiple goals simultaneously - is impaired.
This scenario changes when she faces the reduced set {m1, a1}. Given that the objective of
not appearing as a selfish person to others is stated taking into account the viability of leaving some diversity in the basket - the ‘whenever possible’ part - it does not apply when choice is constrained to the smaller set: since foregoing all fruits is not an option, the diversity of the fruit basket will be reduced regardless of what choice she makes. Thus, when the decision maker faces the smaller set {m1, a1}, the impossibility of leaving two types of fruits in the basket
gives her the ‘wiggle-room’ to behave in accordance with her self-interested preferences with-out worrying abwith-out appearing selfish to others. The opportunity aspect of freedom, therefore, may increase in the smaller set since it gives the individual a functioning previously unattain-able, i.e. the opportunity to pursue actively one of her goals while remaining ‘neutral’ regarding the other, what is presumably preferable to the opportunity of achieving one goal only if inten-tionally abandoning the other altogether - the only possible configuration available when choice is constrained to {m1, a1, a2}. Monotonicity, therefore, is violated in this case.6
In Mary’s decision problem, a similar conclusion may arise. Assume that Mary’s parents have always tried to stimulate that she and Jill should divide their belongings equally with one another, and Mary takes this principle quite seriously. However, at the same time, she treasures her own well-being and the ability to pursue her happiness without always taking note of how others are affected by her choices (although some priority is given to the egalitarian notions that her parents instilled in her and Jill). If her primary goal is associated with ‘whenever possible, sharing equally with Jill’, and her secondary goal is ‘pursuing my own interests’, Mary can experience larger opportunity freedom when she receives only one $100 bill instead of two bills
of $50, i.e., when she faces {x, z} instead of {x, y, z}. As in the ‘mangoes and apples’ example, the larger menu imposes a trade-off between the two goals, while the smaller one allows Mary to behave self-interestedly without deliberately abandoning the egalitarian principle that she also values.
Therefore, in this essay, we elaborate further on the preference-based approach to freedom of choice rankings and propose a rule to rank opportunity sets in terms of freedom when the in-formation of meta-preferences play a role. In our rule, agents have multiple goals and objectives that are related to their plural identities but employ only two of these to assess which options should count to freedom of choice. Each of the two identities, or rationales, is represented by a set of preference relations that, combined with the priority the agent gives to one over another and what relationship she has established with each principle, will constitute the informational basis of our framework. We draw on the rational shortlist method of Manzini and Mariotti (2007), and on procedure β of Houy and Tadenuma (2009) to construct a two-stage rule to identify, for any opportunity sets A and B, the set of relevant alternatives to freedom. The first stage uses the first identity (the first-best set of preferences) to eliminate inferior options within each set, while the second stage uses the second identity (the second-best set of preferences) to eliminate inferior alternatives among all shortlisted options in the first stage. Identified this set for any opportunity sets A and B, our rule ranks A over B if, and only if, the set of short-listed options of A contains more alternatives that can increase freedom of choice than the set of shortlisted options of B.
The role of meta-preferences is twofold. First, preferences over sets of preference relations are used to discard reasonable but dominated rationales. In other words, among all possible reasonable identities, the individual chooses those two that reflect more adequately her under-standing of the state of affairs and what ends she aims to promote in that choice situation. The fact that Mary is, for instance, ‘vegan’, may not be helpful to the problem that she faces even though it might reflect to a large extent the general principles that she takes interest. Second, preferences over the two selected rationales determine the order in which they are used to es-tablish which options should count for freedom, i.e., given two distinct preference orderings R1 and R2, if the agent ranks R1 over R2, then R1 ordering is used at the first stage, and R2
at the second. The order, therefore, matters to compare opportunity sets. We show that this way of assessing alternatives describes an agent that uses choice prescriptions based on R1 as a
sort of ‘inviolable maxim’, but that does not necessarily differentiates a situation where she can directly promote her primary goal (e.g., Mary splits equally the $ with her sister) from another where she simply is unable to violate it (e.g., Mary takes all the money for herself because she cannot exchange a $100 note into two notes of $50).
Thus, we depart from the reasonability approach of Jones and Sugden (1982), Pattanaik and Xu (1998) and Romero-Medina (2001) not only when different weights are given to reasonable preference relations, but also when some of the rankings elicited by reasonable preferences are not used by the agent because they are not useful to the problem in hand, or because they do
not reflect what the agent considers herself to be. Moreover, one key element in our approach is that the value of each option to freedom of choice not only depends on the menu this option originally belongs, but also on the other menu that is used in the comparison. This characteristic makes our rule heavily menu-dependent and, as we show, prone to intransitivity, and also leads to a tension between the intrinsic and the opportunity aspects of freedom of choice: given our focus on the latter and the fact that our rule may fail the monotonicity axiom, it can be argued that the former aspect is not properly depicted in our approach.
Our work, therefore, adds value to the literature by proposing a novel rule to rank oppor-tunity sets in terms of freedom that account for the richer structure of meta-preferences. Even though the literature recognizes that many different reasonable points of view can motivate pref-erence orderings and choices, the consequences of diffpref-erences in the priorities that people give to these principles remained unexplored by the current freedom of choice approaches. More-over, we also provide some new insights on the validity of the monotonicity axiom by arguing that, when a structure that makes room for different priorities is used, the monotonicity prop-erty might be violated and individuals may experience higher freedom when some options are excluded from their opportunity sets.
Apart from this introduction, this article has three more sections. Section 2.2 lay down the notation and axioms. Section Section 2.3 presents the properties of options that are considered to increase freedom of choice. Section 2.4 derives the rule to rank sets in terms of freedom of choice, and discusses the results. Section 2.5 concludes.
2.2 FRAMEWORK
Let X be the universal set of alternatives, assumed to be finite, and let Z = 2X − ∅ be the set of all non-empty subsets of X. A binary relation on X is a set Rk ⊆ X × X, with any pair
(x, y) ∈ Rkread as ‘x is at least as good as y’, and denote by R the set of all reasonable binary
relations on X. Any set Rkis assumed to reflect a different rationale (MANZINI, MARIOTTI,
2007), or identity part (BINDER, 2014) that compose this individual plurality of points of view. Furthermore, individuals have preferences over the rationales that compose their ‘plural-selves’, i.e., the agent’s meta-preferences, where we assume that, for all i, j ∈ {1, . . . , #R}, and all Ri, Rj ∈ R, i < j implies in Ri ranked over Rj according to the individual’s meta-preferences.
We do not assume that each Rk is necessarily complete (for all x, y ∈ X, (x, y) ∈ Rk
or (y, x) ∈ Rk). As observed in Binder (2014), a plurality of reasons often makes room to
incompleteness, either result of the irrelevance of the principle to the comparison in question, or what might be seen as a refusal to rank alternatives when the agent is forced into a ‘Sophie’s choice’ situation.7 Nevertheless, we assume that Rk is transitive (for all x, y, z ∈ X, (x, y) ∈ 7Situations in which individuals are forced to choose between options that will inevitably inflict great damage to others or themselves, or where choosing the less of many evils becomes an extremely difficult decision, have become known as a ‘Sophie’s choice’ because of William Styron’s novel released in 1979 where, Sophie, a Polish refugee, while at Auschwitz had to decide which of her two children would be saved from the concentration camp,
Rk and (y, z) ∈ Rk imply (x, z) ∈ Rk) for k = 2, . . . , #R. Note, therefore, that R1 - the first
and most important principle - is not necessarily transitive.
Moreover, Ri and Rj, for i 6= j, are not necessarily disjoint sets since the criterion used to
construct Ri and Rj do not need to diverge in what relation any two alternatives have. In other
words, for all s, t ∈ X, [(s, t) ∈ Ri] ; [(s, t) 6∈ Rj]. To illustrate, imagine that Ri and Rj
denote preferences elicited by the agent’s ‘football fan’ and ‘sports fan’ identities, respectively, and three possible programs are being evaluated: x = ‘football match’, y = ‘basketball game’, and z = ‘ballet concert’. In this case, while her ‘football fan’ identity would have little to say about the ranking between y and z – such as the ‘sports fan’ part would have little to say about x and y ranking since this identity does not specify which of these two sports is preferred – both identities clearly agree that x must be ranked over z, and that z over x cannot hold. Hence, (x, z) ∈ Ri, and (x, z) ∈ Rj.
Following the notation used in Houy and Tadenuma (2009), for all Rk ∈ R, let P (Rk) =
{(s, t) ∈ Rk| (s, t) ∈ Rk and (t, s) /∈ Rk}, and, for all S ∈ Z, let CRk(S) = {s ∈ S| ∀ t ∈
S, (t, s) /∈ P (Rk)}, for k ∈ {1, . . . , #R}. Hence, CRk(S) is the set of all non-dominated
alternatives in S given Rk. A cycle of preference relations in P (Rk) is a finite sequence (xn)mn=1,
for m ∈ N and m ≥ 2, such that (xn, xn+1) ∈ P (Rk), for all n ∈ {1, . . . , m − 1}, and
(xm, x1) ∈ P (Rk). Moreover, denote by Ck(X) the set of all cycles in X × X for P (Rk), and
let ACk(X) = {x ∈ X| x ∈ (xn)mn=1, for some (xn)mn=1 ∈ Ck(X)}. Any binary relation is
acyclicif it does not lead to a cycle in X. Clearly, assuming that Rk is transitive, for k 6= 1,
imply in Rkbeing acyclic as well.
Let us also state the following definitions that will be helpful to configure our rule.
Definition 1. The choice of x fails with RkinA if x ∈ A and there is a y ∈ A, y 6= x, such that
(y, x) ∈ P (Rk).
Moreover, when an alternative fails with R1 in some set, we say that this option is virtually
unfeasible in that set. The intuition behind this concept is that the first rationale eliminates all options that cannot increase freedom of choice because they have no intrinsic value to the individual’s first - and most important - goal given the opportunity set they belong.
Definition 2. For all A, B ∈ Z, and all alternatives x ∈ A and y ∈ B (with possibly A = B) we say thaty promotes better in B than x in A if, and only if, one of these two cases occur:
a. x fails with R1 inA and y does not fail with R1inB; or
b. neitherx nor y fail with R1 inA and B, respectively, and (y, x) ∈ P (R2).
In other words, given all options in A and B, if x is virtually unfeasible in A while y is not on B, this suffices to ascertain that y promotes better in B than x in A. However, this is a sufficient, but not necessary, condition: it can be the case that both x and y are not virtually
unfeasible. In this case, y promotes better in B (or also in A) than another x in A if y, besides being not dominated in R1 within the set it belongs, R2-dominates x. The menu-dependence is
evident in this scenario: take, for instance, two options x ∈ A and y ∈ B, and assume that both do not fail within their sets considering R1. Therefore, x and y are acceptable choices in each
choice framework. Assume further that choosing x in A and y in B does not fail R2, but the
choice of y in B has the advantage that it is preferable to x: in this case, we may presume that the choice of y in B becomes more valuable than x in A for the sake of promoting the agent’s multiple goals.
Definition 3. For all S, T ∈ Z, let CRiRj(S, T ) = CRj(CRi(S) ∪ CRi(T )).
That is, CRjRi(S, T ) is the set of all non-dominated alternatives given the Rj rationale, in
the union of the Ri non-dominated alternatives in S and T .
Decision makers evaluate freedom of choice of opportunity sets in Z. Denote by% a binary relation over Z, where, for any S, T ∈ Z, S % T means that “S offers at least as much freedom of choice as the opportunity set T ”, with and ∼ denoting, respectively, [S % T and not(T % S)] and [S % T and (T % S)]. Throughout this paper, we assume that, among all possible rationales that the agent can have, she will use only two of them to identify whether – given a comparison between two opportunity sets – one alternative can count to the freedom of choice. Denote by R1 and R2 these two identities, with R1, R2 ∈ R. Based on the works of
Puppe (1996), Pattanaik and Xu (1998) and Puppe and Xu (2010), we assume a related set of conditions that should be respected by our freedom of choice ranking.
Axiom SND (Simple Non-Dominance). For all x, y ∈ X, CR2({x, y}) = {x, y} ⇔ {x} ∼ {y}.
Axiom SND requires that any two alternatives that are not dominated using R2must provide
the same freedom of choice to the agent when compared among each other as opportunity sets. The intuition is that a singleton set cannot fail with R1(or R2, for that matter) because it provides
only one option that will be chosen irrespective of what the individual prefers.
Axiom EX (Expansion). For all A, B ∈ Z, and x ∈ X −A, if CR2R1(A∪{x}, B) = A∪{x}∪B,
thenA % B ⇔ A ∪ {x} B.
Axiom EX determines that, when adding an alternative x to a set A that offers at least as much freedom of choice as another set B, and provided that x is not R1-dominated by A, or
R2-dominated by A’s and B’s R1 non-dominated options, this enlarged set A ∪ {x} must now
be strictly preferable than B in terms of freedom.
Axiom D (Dominance). For all A, B ∈ Z, if CR2R1(A, B) ∩ CR1(B) 6= ∅, then A B.
Axiom D requires that, if a set B has no options in CR2R1(A, B), then it must be dominated
by A provided that CR2R1(A, B) is a non-empty set.
Axiom I, on the other hand, states that A and B must be indifferent in terms of freedom if CR2R1(A, B) is empty.
Axiom COM1 (Composition 1). For all A, B, C, D ∈ Z, such that (A ∩ C) = (B ∩ D) = ∅, andCR2R1(A ∪ C, B ∪ D) = A ∪ B ∪ C ∪ D, then [A % B and C % D ⇔ A ∪ C % B ∪ D.
In order to see the intuition behind COM1 axiom, take four sets A, B, C, D ∈ Z, with A % B and (B ∩ D) = ∅, but assume that A and C have all but one elements in common. If CR2R1(A ∪ C, B ∪ D) = A ∪ B ∪ C ∪ D, regardless the fact that A % B and C % D,
when adding C to A we are increasing by just one the number of options relevant to freedom of choice, while adding D to B might increase substantially the in CR2R1(A ∪ C, B ∪ D) to
the point that it exceeds the share of A ∪ C . Thus, to rule out this possibility, Axiom COM1 requires that (A ∩ C) = (B ∩ D) = ∅.
Axiom COM2 (Composition 2). For all A, B, C, D ∈ Z, such that (A ∩ C) = (B ∩ D) = ∅, andCR2R1(A ∪ C, B ∪ D) = A ∪ B, then A % B ⇔ A ∪ C % B ∪ D.
Axiom COM2, on the other hand, states that freedom comparisons cannot rely on ‘irrel-evant’ alternatives. Thus, for any sets A and B where A is better ranked, this rank does not change when we add dominated options to both sets.
The following properties are desirable considering the intuition behind the freedom of choice approach. The first requires that for every sets A and B there will be at least one al-ternative within these sets that increase freedom of choice, i.e., that is non-dominated given the two stages of elimination. The second is the monotonicity property of Puppe and Xu (2010). Condition NE (Non-Emptiness). For all A, B ∈ Z, CR2R1(A, B) 6= ∅.
Condition M (Monotonicity). For all A, B ∈ Z, if B ⊆ A, then A% B.
Our rule to compare sets in terms of freedom of choice is stated in the definition below. Definition 4. For all A, B ∈ Z,
A %∗ B ⇔ #[CR1(A) ∩ CR2R1(A, B)] ≥ #[CR1(B) ∩ CR2R1(A, B)] (2.1)
2.3 PROPERTIES OF OPTIONS IN CR2R1(A, B)
The following corollary will be helpful to evaluate what properties make options in the set CR2R1(A, B) suited for our analysis.
Corollary 1. For all x ∈ CR2R1(A, B), if x fail with R1 forA, then x does not fail with R1 for
Corollary 1 states that any option in CR2R1(A, B) cannot fail with the first rationale in both
A and B simultaneously. Take a x ∈ CR2R1(A, B), where x 6∈ (A ∩ B). By Definition 1, it is
obvious that x cannot fail in B if x does not belong to B, but it is also clear that x does not fail with R1 in A otherwise it would be impossible to have x ∈ CR2R1(A, B). By the same token,
if x ∈ (A ∩ B) and fails with R1 in A and B, then x 6∈ CR2R1(A, B). Therefore, for at least
one set, any x ∈ CR2R1(A, B) is not virtually unfeasible and can cope with the first goal that
the individual has put to herself.
Nevertheless, alternatives in CR2R1(A, B) can fail with R2 in both A and B. Lemma 1
denotes in what conditions this may occur.
Lemma 1. Let x ∈ CR2R1(A, B) such that x ∈ (A ∩ B). If there is a y ∈ A and z ∈ B (with
the possibility thaty = z) such that {(y, x), (z, x)} ⊆ P (R2), then y fails with R1 inA and z
fails withR1inB.
Proof. Assume, to the contrary, that y does not fail in A. Then, y ∈ CR1(A) ∪ CR1(B), so the
fact that (y, x) ∈ P (R2) imply in x 6∈ CR2R1(A, B), what is a contradiction. Thus, y must fail
in A and, by the same reasoning, z must fail in B with R1, completing the proof.
By Lemma 1, we have that alternatives in CR2R1(A, B) may fail with R2 for A, or B, or
both, only when we take into account virtually unfeasible alternatives, i.e., options that are R1
-dominated. Intuitively, this agent believes that choosing a suboptimal alternative given R2 is
justifiable for the sake of not infringing R1.
Lemma 2. For all y ∈ (A ∪ B − CR2R1(A, B)), there is an x ∈ CR2R1(A, B) such that x
promotes better inA or B than y in A or B.
Proof. If y ∈ (A ∪ B − CR2R1(A, B)), then it might be the case that y is virtually unfeasible
in, say, A. Thus, by Definition 2, any option in x ∈ CR2R1(A, B) promotes better in A or
B than y in A. Now, assume that y is not virtually unfeasible in A, and assume that none x ∈ CR2R1(A, B) promotes better than y. Therefore, y ∈ CR1(A) ∪ CR1(B), and no x ∈
CR2R1(A, B) is such that (x, y) ∈ P (R2). But if that is true, then y ∈ CR2R1(A, B), what is a
contradiction. Therefore, there must exist a x ∈ CR2R1(A, B) that promotes better in A or B
than y in A.
Example 1: Let X = {b, c, f, g, i, s}, A = {b, c, f } and B = {c, g, i, s}, and assume that the first rationale elicits the set
P (R1) = {(b, c), (f, c), (g, c), (i, c), (s, c), (f, g)}
what yields CR1(A) = {b, f } and CR1(B) = {g, i, s}. Now, assume that the preference
P (R2) = {(c, b), (c, g), (g, s), (c, s), (s, f ), (g, f )}
Clearly, CR1(A) ∪ CR1(B) = {b, f, g, i, s}, but
CR2R1({b, f, g, i, s}) = {b, g, i}
Why options c, f and s are not considered to improve opportunity freedom when we compare A and B? Take first the option c: it is dominated in A and B given R1, so c is virtually unfeasible
in both sets. Therefore, c is ruled out as an option that counts to the comparison of A and B in terms of freedom.
Given the shortlisted options, the individual follows to the second stage. Let us analyze CR1(A) = {b, f }. Note that neither b dominates f , nor f dominates b, meaning that, when
having to make a choice in A, both options are perfectly acceptable since f does not fail with R1 and R2, while b failure with R2 is due to the presence of a virtually unfeasible option in A.
However, our rule does not evaluate A in isolation: given the comparative exercise that makes options in A be assessed taking into account information brought by B, the agent realizes that f is dominated by g, and even though g is only feasible in B, the choice of g in B promotes better her goals and objectives than the choice of f in A. When analyzing CR1(B) = {g, i, s}
in the light of the second rationale, we have that s is dominated by g, meaning that s fails with R2 in B.
Thus, for all x ∈ CR2R1(A, B):
a. x ∈ CR2R1(A, B) does not fail with R1 for A and B simultaneously.
b. If x fails with R2 in A because, say, there is a y ∈ A such that (y, x) ∈ P (R2), then y
fails with R1 in A. So y is virtually unfeasible, what gives one the “wiggle room” to fail
the second principle. This is the case of g in B: it fails with R2 in B only because the
virtually unfeasible option c.
c. Take any y ∈ (A ∪ B − CR2R1(A, B)). Then, there is a x ∈ CR2R1(A, B) such that
(a) If x ∈ A and y ∈ A, the choice of x in A promotes better than the choice of y in A. Since c is virtually unfeasible, by definition b, g and i must promote better than c in A.
(b) If x ∈ A and y ∈ B, the choice of x in A promotes better than the choice of y in B. Take f ∈ A: clearly g promotes better in B than f promotes in A.
Example 2: Assume that an individual is deciding where she will spend her vacations. Op-tions are b =‘Buenos Aires’, p =‘Paris’, r =‘Rio de Janeiro’, s =‘Santiago de Chile’, and
t =‘Tokyo’. All opportunities are given in the set X = {b, p, r, s, t}. The set R1 is motivated
by ‘traveling to South America’, and R2 is ‘go to a wine-tasting tour’. Let A = {b, p, s, t}, and
B = {p, t}. In this scenario, we have CR1(A) = {b, s}, and CR1(B) = {p, t}. Note that, given
that neither Paris nor Tokyo are South American cities, we cannot say that they are virtually unfeasible in the set B: however, Paris and Tokyo are virtually unfeasible in the set A since they are R1-dominated by b and s.
Going to the second stage, we have CR1(A) ∪ CR1(B) = {b, p, s, t}. Suppose that P (R2) =
{(b, t), (p, t), (s, t), (b, r), (p, r), (s, r)}. In other words, this agent believes that in Rio and Tokyo she will not be able to take a wine-tasting tour. Thus, CR2R1(A, B) = {b, p, s}. For
this configuration of A and B, going to Paris promotes better than going to Tokyo in set B, and both Buenos Aires and Santiago promote better in A than Tokyo in B.
However, assume a different set of preferences given by P (R2) = {(b, r), (p, r), (s, r)}.
In this case, CR2R1(A, B) = {b, p, s, t} and, if freedom of choice is assessed through (4), we
would have A indifferent to B. Nonetheless, this conclusion may be seen as counterintuitive within the idea of promoting distinct goals and if we take note that, in A, the decision maker can effectively choose an option that is both a south american city and has wine-tasting tours, while in B she just does not violate R1. This example highlights the fact that, regarding R1,
the individual that ranks sets according to (4) is indifferent between promoting it directly or just not violating it deliberately. That is the importance of the whenever possible expression: our agent puts as her first principle only an objective that can be stated in relative, and not absolute terms. The first objective, therefore, is inviolable, as a ‘maxim’, and this suffices to our agent as if she dislikes behaving in a way that is in disagreement with what is expected given R1. As
long as two different opportunity sets provide the same number of good alternatives that do not violate R1, it does not matter whether in one she has the possibility of actively accomplishing
some situation in line with R1while in the other she simply does not violate it.
2.4 RESULTS
In this section, we present our results given the axioms proposed in Section 2.2. The fol-lowing lemma will be helpful to state the proof of the main proposition.
Lemma 3. If s ∈ T ⊂ S, [s ∈ CRk(S)] ⇒ [s ∈ CRk(T )].
Proof. Suppose s ∈ T ⊂ S, such that s ∈ CRk(S), but s 6∈ CRk(T ). Thus, there is a w ∈ T
such that (w, s) ∈ P (Rk). Since T ⊂ S, w ∈ S, and, consequently, s 6∈ CRk(S), contradicting
our first assumption. Therefore, [s ∈ CRk(S)] ⇒ [s ∈ CRk(T )].
Proposition 1. % satisfies Axioms SND, EX, D, I, COM1 and COM2 iff %=%∗
Proof. (This proof is based on a similar reasoning used in Pattanaik and Xu (1998)). We only prove sufficiency. Suppose CR1(A) ∩ CR2R1(A, B) = {¯a1, . . . , ¯am}, CR1(B) ∩ CR2R1(A, B) =
{¯b1, . . . , ¯bn}. First, let m = n > 0. Then, A, B ∈ Z are two sets such that #[CR1(A) ∩
CR2R1(A, B)] = #[CR1(B) ∩ CR2R1(A, B)] = n.
Since, for any x ∈ X, CR1({x}) = {x}, it must hold that CR2R1({¯a1}, {¯b1}) = CR2({¯a1, ¯b1}).
By Lemma 3, given {¯a1, ¯b1} ⊂ CR2R1(A, B), we have CR2R1({¯a1}, {¯b1}) = {¯a1, ¯b1}. Using
Axiom SND,
{¯a1} ∼ {¯b1} (2.2)
In the same way,
{¯a2} ∼ {¯b2} (2.3)
Given that (¯a1∩ ¯a2) = (¯b1∩¯b2) = ∅, and CR2R1({¯a1, ¯a2}, {¯b1, ¯b2}) = {¯a1, ¯a2, ¯b1, ¯b2}, by Axiom
COM1,
{¯a1, ¯a2} ∼ {¯b1, ¯b2} (2.4)
Repeating the use of Axioms SND and COM1 for ¯a3, ¯b3, . . . , ¯an, ¯bn, we achieve
{¯a1, . . . , ¯an} ∼ {¯b1, . . . , ¯bn} (2.5)
Let A = {{¯a1, . . . , ¯an} ∪ A0} and B = {{¯b1, . . . , ¯bn} ∪ B0}. Clearly, CR2R1({{¯a1, . . . , ¯an} ∪
A0, {¯b1, . . . , ¯bn} ∪ B0) = CR2R1(A, B). Thus, using Axiom COM2, we achieve
A ∼ B (2.6)
Now, assume that CR2R1(A, B) = ∅. Clearly we have CR1(A) ∩ CR2R1(A, B) = CR1(B) ∩
CR2R1(A, B) = ∅. Therefore, A, B ∈ Z are two sets such that #[CR1(A) ∩ CR2R1(A, B)] =
#[CR1(B) ∩ CR2R1(A, B)] = 0. Using Axiom I, we have A ∼ B.
Let us prove #[CR1(A) ∩ CR2R1(A, B)] > #[CR1(B) ∩ CR2R1(A, B)] ⇒ A B. First,
assume n = 0 and CR2R1(A, B) 6= ∅, what imply in m > 0. Direct use of Axiom D implies
A B.
Now, assume m > n > 0. Take any ¯ai ∈ {¯an+1, . . . , ¯am}. Clearly, ¯ai ∈ CR2R1({¯a1, . . . , ¯an}
∪{ ¯ai} ∪ {¯b1, . . . , ¯bn}). Using (2.5) and Axiom EX,
{¯a1, . . . , ¯an} ∪ {¯ai} {¯b1, . . . , ¯bn} (2.7)
Repeating this process for the (m − n − 1) alternatives in {an+1, . . . , ¯am} − {ai}, we have,
{¯a1, . . . , ¯an} ∪ {¯an+1, . . . , ¯am} {¯b1, . . . , ¯bn} (2.8)
Now, let A = {{¯a1, . . . , ¯an} ∪ {¯an+1, . . . , ¯am} ∪ A00}. Using A00, B0, (2.8) and Axiom COM2,
and, finally,
A B (2.10)
and completes the proof.
It is important to remark that our rule does not repeat the same results that would be achieved using the rule based on reasonable preferences proposed in Pattanaik and Xu (1998). To make this distinction clear, let X = {x, y, a, w, z}, A = {x, w, z}, and B = {y, a}, and preferences
R1 = {(x, a), (x, y), (x, w)
(y, a), (y, x), (y, w) (z, a), (z, x), (z, y), (z, w) (w, a), (w, x), (w, y), (w, z)} and
R2 = {(x, a), (x, y), (x, z), (x, w)
(y, a), (y, x), (y, z), (y, w) (a, x), (a, z), (a, w) (z, w)
(w, z)}
Therefore, we have P (R1) = {(x, a), (y, a), (w, a), (z, a), (z, x), (z, y)} and P (R2) = {(x, z),
(x, w), (y, a), (y, z), (y, w), (a, z), (a, w)}. However, if we do not distinguish between R1 and
R2, and attribute the same weight to both sets, the set of interest to identify non-dominated
alternatives would be P (R1 ∪ R2) = {(y, a)}. Using one of the rules proposed in Pattanaik
and Xu (1998) to compare sets A and B in terms of freedom of choice, we should identify all maximal elements in A that are not dominated by options of B, and all maximal elements of B not dominated by alternatives in A (denoted by max(A) − ABand max(B) − BA, respectively), considering all reasonable preference relations. In this case, the only dominance relation that can be reached is between y and a: whenever option y is available, a is dominated by y, and clearly a /∈ max(B). Thus,
max(A) − AB = {x, w, z} (2.11)
and
max(B) − BA= {y} (2.12)
Comparing the cardinalities of (2.11) and (2.12), one may conclude that, using Pattanaik and Xu (1998) rule, A B. Now, let us apply our rule proposed in equation (2.1). In this case,
CR1(A) = {w, z}, CR1(B) = {y}, and CR2R1(A, B) = CR2({w, z, y}) = {y}. Hence,
CR1(A) ∩ CR2R1(A, B) = ∅
and
CR1(B) ∩ CR2R1(A, B) = {y}
Using our rule, we have B A.
This difference has some important implications. As already stressed, the individual that ranks opportunity sets in line with our rule regards freedom to be dependent not only on pref-erences that could be reasonably adopted, but also on his judgments about how each of these preferences reflects his plural identity. Take, for instance, the beheaded at dawn example pro-posed in Sen (1993b): at first glance, one should expect the addition of such terrible alternative to the opportunity set to have no impact in the freedom enjoyed by the agent. Pattanaik and Xu (1998), however, make room for the possibility that even this undesirable option could eventually increase freedom: if ‘spending 50 years in a solitary cell’ was the only alternative at disposal, enlarging the set by ‘beheaded at dawn’ could increase freedom of choice since choosing to be beheaded, instead of spending a lifetime in prison, does not sound completely absurd anymore. Nonetheless, even casting no doubt on the reasonability of this choice in such a desperate situation, this agent can be strongly ‘pro-life irrespective of the circumstances’ with this dimension reflecting an important part of her plural identity. In such a scenario, offering an alternative as ‘beheaded at dawn’ would hardly improve her freedom since it fails to meet the demands imposed by this part of the agent’s identity.
We now investigate under which conditions CR2R1(A, B) satisfies Condition NE. This is
par-ticularly important for set comparisons since CR2R1(A, B) indicates which alternatives, given
the exercise of comparing A and B, increase freedom. So, to say that CR2R1(A, B) can be
empty amounts to say that there may be a situation where neither A nor B, when compared to each other, are able to provide this agent any minimal standard of freedom of choice, what is an apparently counterintuitive conclusion. The following corollary will be useful to state our results.
Corollary 2. Let j = 1, 2. For all A ∈ Z, CRj(A) 6= ∅ if and only if P (Rj) is acyclic,
Clearly, one possible way to CR2R1(A, B) violate Condition NE is when CR1(A) ∪ CR1(B)
is an empty set. To ensure that at least one alternative passes to the second round of elimination, we introduce the following properties that impose some domain restrictions to P (R1).
Condition 1. #C1(X) ≤ 1.
Condition 2. For all z ∈ (X − AC1(X)), and for all x ∈ AC1(X), (x, z) /∈ P (R1).
Condition 1 states that there is at most one sequence of options that is a cycle for P (R1).
case, P (R1) violate Condition 1 since there are two cycles, denoted by sequences (x, y, z, w),
and (x, y, z). Condition 2 proposes that any alternative within a cycle must be dominated by the options without cycles for preferences given by P (R1).
Lemma 4 shows that, under Conditions 1 and 2, the first round of elimination of inferior alternatives must result in a non-empty set.
Lemma 4. For all A, B ∈ Z, with A 6= B, (CR1(A) ∪ CR1(B)) 6= ∅ if and only P (R1) satisfies
Conditions 1 and 2.
Proof. (Sufficiency). Suppose #C1(X) ≤ 1 (Condition 1), and, for all z ∈ (X − AC1(X)),
and for all x ∈ AC1(X), (x, z) /∈ P (R1) (Condition 2). We must show that this imply in
(CR1(A) ∪ CR1(B)) 6= ∅, for all A, B ∈ Z, with A 6= B. When #C1(X) = 0, P (R1) is acyclic
and this suffices to both CR1(A) and CR1(B) be non-empty, culminating in the result. Now, let
#C1(X) = 1. Hence, there is only one cycle (xn)mn=1 for P (R1), and let S1 be the set of all
alternatives in this cycle. Clearly, CR1(S1) = ∅.
Assume, to the contrary, that both Conditions 1 and 2 are satisfied, but we may have (CR1(A) ∪ CR1(B)) = ∅, with A 6= B. It is obvious that CR1(A) = CR1(B) = ∅. By
Corollary 1, and using the fact that #C1(X) = 1, S1 must lie within A and B, and we can
rewrite both sets as A = S1∪ V and B = S1 ∪ U , with V, U any (not necessarily non-empty)
subsets of X. Let V = U = ∅. So, CR1(A) = CR1(B) = ∅, but A = B, contradicting the
assumption that A 6= B. Hence, we must have V 6= ∅, and/or U 6= ∅. Let V 6= ∅. Thus, there is at least one alternative z ∈ A such that z /∈ S1. Using Condition 1, we know that there is
no cycle in V , and, by Condition 2, z is non-dominated by any x ∈ S1. Thus, we must have
CR1(A) 6= ∅, implying in (CR1(A) ∪ CR1(B)) 6= ∅, which contradicts our initial assumption,
and completes the sufficiency part of the proof.
(Necessity).Assume that, for all A, B ∈ Z, with A 6= B, we have (CR1(A) ∪ CR1(B)) 6= ∅. We
must show that Conditions 1 and 2 hold. First, assume, to the contrary, that Condition 1 does not hold. Hence, there can be more than one cycle in X for P (R1), and let #C1(X) = q > 1 ∈ N,
with S1, . . . , Sq ∈ Z denoting the alternatives within these q distinct cycles.8 Making A = Si
and B = Sj, for any i, j ∈ {1, . . . , q}, and i 6= j, by Corollary 2 we have CR1(A) = CR1(B) =
∅, leading to a contradiction. Thus, [(CR1(A) ∪ CR1(B)) 6= ∅] ⇒ #C1(X) ≤ 1.
Now, assume that Condition 2 does not hold, but for all A, B ∈ Z, with A 6= B, we have (CR1(A) ∪ CR1(B)) 6= ∅. Therefore, there is a z ∈ (X − AC(X)) and a x ∈ AC(X), with
(x, z) ∈ P (R1). Without loss of generality, let S1, S2 ∈ Z denote two sets of alternatives
within a cyclical sequence (S1 and S2 may be equal), with x ∈ S1. Let A = S1 ∪ {z}, and
B = S2. Clearly, z /∈ CR1(A), and (CR1(A) ∪ CR1(B) = ∅, which is a contradiction. So,
[(CR1(A) ∪ CR1(B)) 6= ∅] ⇒ [For all z ∈ (X − AC1(X)), and for all x ∈ AC1(X), (x, z) /∈
P (R1)], and completes the necessity part of the proof. 8Note that S
1may be a subset of S2, i.e., S1alternatives constitute a subcycle of S2, and so on. Also, AC1(X) = S1∪ . . . ∪ Sq.
Hence, any pair (A, B) ∈ Z × Z is able to offer some alternatives to the second stage of freedom of choice assessment only when the existence of cycles in preferences elicited by the first rationale is limited to only one sequence of alternatives. Also, any option outside this cy-cle must dominate ‘within-cycy-cle’ options. Though it weakens the necessity of acyclicity for P (R1), the existence of cycles can be managed only under specific situations. For instance,
suppose that there is a cycle in preferences over {car, minivan, bus}, and the only remaining alternative is ‘executive jet’. Assuming that R1 denotes the individual preferences when the
social status of alternatives during a thousand miles trip is taken into account, it seems undeni-able that a luxurious executive jet would win any other alternative by a landslide. In this case, any set composed by {car, minivan, bus, executive jet} would succeed to provide candidates to the second round appraisal. Nevertheless, the same conclusion would hardly be maintained if ‘executive jet’ were substituted by ‘bicycle’: in this scenario, any alternative that composes the cycle apparently provides higher social status, making ‘bicycle’ a dominated option.
Given Lemmas 3 and 4, together with our assumption that R2 is transitive we ensure that
CR2R1(A, B) is non-empty. However, allowing R2to be intransitive, Proposition 2 states under
which conditions CR2R1(A, B) will satisfy Condition NE.
Proposition 2. Assume that R2 does not satisfies transitivity. For allA, B ∈ Z, with A 6= B,
CR2R1(A, B) satisfies Condition NE if and only if (CR1(A) ∪ CR1(B)) 6= ∅, and, for every cycle
(xn)nn=1forP (R2), there is an alternative x ∈ AC2(X) such that x /∈ (CR1(A) ∪ CR1(B)).
Proof. (Sufficiency). Assume that CR2R1(A, B) satisfies Axiom NE. Clearly, either CR1(A)
and/or CR1(B) are non-empty. Now, assume there is a cycle for P (R2), and denote alternatives
in this cycle by S. Letting CR1(A) ∪ CR1(B) = S, it is clear that CR2(S) = ∅, contradicting our
first assumption. Hence, there is at least one alternative x ∈ S such that x /∈ (CR1(A)∪CR1(B))
for this cycle, what completes the sufficiency part of the proof.
(Necessity). It is immediate to see that [(CR1(A) ∪ CR1(B)) = ∅] ⇒ [CR2R1 (A, B) = ∅, and,
thus, we must have a non-empty set in (CR1(A) ∪ CR1(B)). Now, assume that, for every cycle
(xn)nn=1for P (R2), there is an alternative x ∈ AC2(X) such that x /∈ (CR1(A) ∪ CR1(B)), and
denote the set of alternatives in the j−th cycle on X by Sj. If this condition holds, (CR1(A) ∪
CR1(B)) 6= Sj, for all j ∈ {1, . . . , q}. Thus, by Corollary 1, CR2(CR1(A) ∪ CR1(B)) 6= ∅, and
completes the necessity part of the proof.
Also, it is worth noting that the lexicographic procedure β (HOUY, TADENUMA, 2009) with two ‘elimination stages’, when applied separately to each set in order to construct a set of relevant alternatives to freedom, will not necessarily yield the set CR2R1(A, B). This strategy
